Solve for x x^(2/5)+x^(1/5)+1=3

$x^{\frac{2}{5}} + x^{\frac{1}{5}} + 1 = 3$

Step 1

Find a common factor $x^{\frac{1}{5}}$ that is present in each term.

${(x^{\frac{1}{5}})}^{2} + x^{\frac{1}{5}} + 1 - 3$

Step 2

Substitute $u$ for $x^{\frac{1}{5}}$ .

${(u)}^{2} + u + 1 - 3 = 0$

Step 3

Solve for $u$ .

Subtract $3$ from $1$ .

$u^{2} + u - 2 = 0$

Factor $u^{2} + u - 2$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Write the factored form using these integers.

$(u - 1) (u + 2) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u - 1 = 0$

$u + 2 = 0$

Set $u - 1$ equal to $0$ and solve for $u$ .

Set $u - 1$ equal to $0$ .

$u - 1 = 0$

Add $1$ to both sides of the equation.

$u = 1$

Set $u + 2$ equal to $0$ and solve for $u$ .

Set $u + 2$ equal to $0$ .

$u + 2 = 0$

Subtract $2$ from both sides of the equation.

$u = - 2$

The final solution is all the values that make $(u - 1) (u + 2) = 0$ true.

$u = 1, - 2$

Step 4

Substitute $x$ for $u$ .

$x^{\frac{1}{5}} = 1, - 2$

Step 5

Solve for $x^{\frac{1}{5}} = 1$ for $x$ .

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{5}})}^{5} = 1^{5}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{1}{5}})}^{5}$ .

Multiply the exponents in ${(x^{\frac{1}{5}})}^{5}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{5} \cdot 5} = 1^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$x^{\frac{1}{5} \cdot 5} = 1^{5}$

Rewrite the expression.

$x^{1} = 1^{5}$

Simplify.

$x = 1^{5}$

Simplify the right side.

One to any power is one.

$x = 1$

Step 6

Solve for $x^{\frac{1}{5}} = - 2$ for $x$ .

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{5}})}^{5} = {(- 2)}^{5}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{1}{5}})}^{5}$ .

Multiply the exponents in ${(x^{\frac{1}{5}})}^{5}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{5} \cdot 5} = {(- 2)}^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$x^{\frac{1}{5} \cdot 5} = {(- 2)}^{5}$

Rewrite the expression.

$x^{1} = {(- 2)}^{5}$

Simplify.

$x = {(- 2)}^{5}$

Simplify the right side.

Raise $- 2$ to the power of $5$ .

$x = - 32$

Step 7

List all of the solutions.

$x = 1, - 32$

Solve for x x^(2/5)+x^(1/5)+1=3

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